Ordinal efficiency under the lens of duality theory

Athanassoglou, Stergios (2010): Ordinal efficiency under the lens of duality theory.

Abstract

An allocation's ordinal efficiency deficit (OED) is defined as the greatest ordinal efficiency loss that can result from its application. More precisely, an allocation's OED is the negative of the greatest total amount by which it may be stochastically dominated by another feasible allocation. Thus, an allocation is ordinally efficient if and only if its OED is zero. Using this insight, we set up a linear program whose optimal objective value corresponds to a given allocation's OED. Furthermore, we show that the OED is a piecewise-linear convex function on the set of allocations. We use the optimal dual variables of the linear program to construct a profile of von Neumann-Morgenstern (vNM) utilities that is compatible with the underlying ordinal preferences, and which is a subgradient of the OED at the given allocation. When the given allocation is ordinally efficient, our analysis implies that it is ex-ante welfare maximizing at the constructed vNM profile, and we recover the ordinal efficiency theorem due to McLennan (2002)

• Item Type: MPRA Paper
• Language: English
• Keywords: random assignment; ordinal efficiency; linear programming; duality
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis; D - Microeconomics > D0 - General > D01 - Microeconomic Behavior: Underlying Principles; D - Microeconomics > D6 - Welfare Economics > D60 - General

References

A. Abdulkadiroglu and T. Sonmez (2003), ``Ordinal Efficiency and Dominated Sets of Assignments,'' {\em Journal of Economic Theory}, 112, 157--172.

D. Bertsimas and J. Tsitsiklis (1997), {\em Introduction to Linear Optimization}, Athena Scientific.

A. Bogomolnaia and H. Moulin (2001), ``A New Solution to the Random Assignment Problem,'' {\em Journal of Economic Theory}, 100, 295-328.

Y.-K. Che and F. Kojima (2009), ``Asymptotic Equivalence of Probabilistic Serial and Random Priority Mechanisms,'' {\em Econometrica}, forthcoming.

A. K. Katta and J. Sethuraman (2006), ``A Solution to the Random Assignment Problem on the Full Preference Domain,'' {\em Journal of Economic Theory}, 131, 231--250.

O. Kesten (2009), ``Why Do Popular Mechanisms Lack Efficiency in Random Environments?'' {\em Journal of Economic Theory}, 144, 2209--2226.

M. Manea (2008), ``Random Serial Dictatorship and Ordinally Efficient Contracts,'' {\em International Journal of Game Theory}, 36, 489--496.

M. Manea (2008), ``A Constructive Proof of the Ordinal Efficiency Theorem,'' {\em Journal of Economic Theory}, 141, 276--281.

M. Manea (2009), ``Asymptotic Ordinal Inefficiency of Random Serial Dictatorship,'' {\em Theoretical Economics}, 4, 165--197.

A. McLennan (2002), ``Ordinal efficiency and the Polyhedral Separating Hyperplane Theorem,'' {\em Journal of Economic Theory}, 105, 435--449.